The problem asks to simplify two rational expressions. Expression (e): $\frac{2x^2 + 17x + 21}{(x+2)(x^2-9)}$ Expression (i): $\frac{3x - 10x - 24}{x(x^2-4)}$

1. Problem Description

The problem asks to simplify two rational expressions.

Expression (e): $\frac{2x^2 + 17x + 21}{(x+2)(x^2-9)}$

Expression (i): $\frac{3x - 10x - 24}{x(x^2-4)}$

2. Solution Steps

(e)

First, factor the numerator $2x^2 + 17x + 21$.

We are looking for two numbers that multiply to $2 \cdot 21 = 42$ and add up to $17$. These numbers are $3$ and $14$.

So we can rewrite the quadratic as $2x^2 + 3x + 14x + 21$.

Factor by grouping: $x(2x+3) + 7(2x+3) = (2x+3)(x+7)$.

Thus, $2x^2 + 17x + 21 = (2x+3)(x+7)$.

Now, factor the denominator.

$x^2 - 9 = (x-3)(x+3)$, so the denominator is $(x+2)(x-3)(x+3)$.

The expression becomes $\frac{(2x+3)(x+7)}{(x+2)(x-3)(x+3)}$.

There are no common factors between the numerator and the denominator, so the expression is simplified.

(i)

First, simplify the numerator: $3x - 10x - 24 = -7x - 24$.

Next, factor the denominator.

$x^2 - 4 = (x-2)(x+2)$, so the denominator is $x(x-2)(x+2)$.

The expression becomes $\frac{-7x-24}{x(x-2)(x+2)}$.

There are no common factors between the numerator and the denominator, so the expression is simplified.

3. Final Answer

(e) $\frac{(2x+3)(x+7)}{(x+2)(x-3)(x+3)}$

(i) $\frac{-7x-24}{x(x-2)(x+2)}$